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Number Theory Level 2

An artist is planning on mixing together any number of different colors from her palette. A mixture results as long as the artist combines at least two colors. If the number of possible mixtures is less than 500, what is the greatest number of colors the artist could have in her palette?

9 10 11 8

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Abdulrahman El Shafei by Abdulrahman El Shafei , 23, Egypt

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3 solutions

Yago Vaz
Nov 27, 2014

If you can mix any number of colors and you have n colors to choose x of them to mix. So, if you sum from x=2 to x=n you have (2^(n))-(n+1) <500

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Bk Lim
Dec 18, 2014

nC0 + nC1 + nC2 + ... + nCn = 2^n

2^9 = 512

512 - 9C0 - 9C1 = 502 > 500

n max = 8

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Anatoliy Razin
Nov 23, 2014

[ l o g 2 500 ] [log_2 500]

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